Is it permissible for a woman who has been beaten by her husband to the point that she has been hospitalized to seek a Khul'ah?

Yes it is permissible for her to do so because this situation constitutes bad and harmful living conditions. She would give back the dowry and be released from the marriage by way of the Khul'ah. Rather, if she was in a place where there's an Islamic Judge, the judge would dissolve the marriage due to this abuse...